Title: Counting associatives and Seiberg-Witten equations

Date: 10/02/2017

Time: 4:10 PM - 5:30 PM

Place: C304 Wells Hall

There is a natural functional on the space of orientation 3-dimensional submanifolds in a G2-manifold. Its critical points are associative submanifolds, a special class of volume-minimizing submanifolds which obey an elliptic deformation theory. Given this, it is a natural question whether one can count associative submanifolds in order to construct an enumerative invariant for G2–manifolds. I will explain several geometric scenarios, which prohibit a naive count of such submanifolds cannot possible be invariant. I will then go on to discuss how (generalized) Seiberg-Witten equations might help cure these problems.

Speaker: Akos Nagy, Fields Institute/University of Waterloo

Title: BPS monopoles with non-maximal symmetry breaking and the Nahm transform

Date: 10/03/2017

Time: 11:00 AM - 12:00 PM

Place: C304 Wells Hall

The notion of “broken symmetry” is central in gauge theories. For BPS monopoles, symmetry breaking can be defined in terms of the eigenvalues of the Higgs-field at infinity. The symmetry breaking is maximal if the eigenvalues are distinct. Monopoles with maximal symmetry breaking have been studied extensively by both mathematicians and physicists for decades now, but little is known about the general case.
In this talk, I will show how to produce monopoles with arbitrary symmetry breaking using the Nahm transform, and I will also outline the construction of its inverse. The inversion heavily uses first order elliptic PDE's on non-compact spaces, more concretely, the theory of Callias-type operators in 3D.
This is a joint project with Benoit Charbonneau.

Schur functions are among the most useful bases for symmetric functions, but they come at the cost of making certain computations much less obvious than when done in other bases. In particular, how can we determine the coefficients of a product of Schur functions in the basis of Schur functions? There are many equivalent ways to computing these numbers; this talk will discuss jeu de taqin, which is an equivalence among skew tableaux, and apply it to obtain a formulation of the Littlewood-Richardson rule, which answers our question. If time allows, we will discuss other important formulations of this rule.

Title: The Development of Shocks in Compressible Fluids

Date: 10/04/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

The lecture shall trace the history of the theoretical study of the formation and evolution of shocks in compressible fluids, starting with the fundamental work of Riemann, the first work on nonlinear hyperbolic partial differential equations. Riemann considered the case of plane symmetry where the problem reduces to 1 spatial dimension. One milestone in the development of the theory was the work of Sideris who gave the first general proof of the finite time breakdown of smooth solutions in 3 spatial dimensions. Another milestone was the work of Majda who first addressed the problem of the local in time continuation of a shock front as a nonlinear free boundary problem for a nonlinear hyperbolic system of partial differential equations. I shall then discuss my own work, which uses differential geometric methods and resolves the resulting singularities giving a complete description in terms of smooth functions.
My first work studies the maximal smooth development of given smooth initial data, the boundary of the domain of this development, and the behavior of the solution at this boundary. The boundary contains certain singular hypersurfaces which originate from certain singular surfaces. The singular surfaces do occur in nature, but not the singular hypersurfaces. My second work studies the physical evolution beyond the singular surfaces by solving a nonlinear free boundary problem with singular initial conditions associated to each of the singular surfaces. From each singular surface a shock hypersurface issues which appears as the corresponding free boundary.

I will describe how cluster algebras arise in hyperbolic geometry of Riemann surfaces \Sigma_{g,s} with s>0 holes with the identification of cluster variables with Penner's lambda lengths, X-variables with shear coordinates, and terms of exchange matrices --- with coefficients of Poisson relations between the shear coordinates. I also describe sets of geodesic functions and their algebras induced by semiclassical/quantum relations for the shear coordinates.

In the study of quantum phases, the concept of topological invariant has emerged as a new paradigm beyond that of Landau theory. The relevance of topology for the classification of phases has been known since the discovery of the quantum hall effect. However, recent theoretical and experimental discoveries of new topological insulators has led to a renewed interest. The purpose of this reading group is to explore both recent and classical results for topological insulators including but not limited to (1) bulk-boundary correspondence (2) K-theoretic classification of topological insulators (3) topological invariants in the presence of disorder (4) quantization of Hall conductance in interacting systems.

Title: Wall-crossing for the Seiberg-Witten equation with two spinors

Date: 10/05/2017

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

Unlike for the classical Seiberg-Witten equation, compactness fails for the Seiberg-Witten equation with multiple spinors. This non-compactness is caused by Fueter sections with values in the moduli space of charge 1 SU(n) ASD instantons. In the simplest case, n = 2, those are Z/2 harmonic spinors. In this talk I will explain in more detail what the preceding sentences mean and then discuss the wall-crossing caused by the appearance of (non-singular) Z/2 harmonic spinors. Time permitting, I will discuss how our wall-crossing formulae can be used to prove the existence of singular Fueter sections.
This is joint work with Aleksander Doan.

Defects are interfaces that mediate between two wave trains with possibly different wave numbers. Of particular interest in applications are sources for which the group velocities of the wave trains to either side of the defect point away from the interface. While sources are ubiquitous in experiments and can be found easily in numerical simulations of appropriate models, their stability analysis still presents many challenges. One difficulty is that sources are not travelling waves but are time-periodic in an appropriate moving coordinate frame. A second difficulty is that perturbations are transported towards infinity, which makes it difficult to apply various commonly used approaches. In this talk, I will discuss nonlinear stability results for sources in general reaction-diffusion system and outline a proof that utilizes pointwise estimates.

We will define a Gröbner basis which is a particular nice generating set for an ideal that has applications in solving polynomial equations, testing ideal membership, and elimination theory.

Title: Counting associatives and Seiberg-Witten equations (2)

Date: 10/09/2017

Time: 4:10 PM - 5:30 PM

Place: C304 Wells Hall

There is a natural functional on the space of orientation 3-dimensional submanifolds in a G2-manifold. Its critical points are associative submanifolds, a special class of volume-minimizing submanifolds which obey an elliptic deformation theory. Given this, it is a natural question whether one can count associative submanifolds in order to construct an enumerative invariant for G2–manifolds. I will explain several geometric scenarios, which prohibit a naive count of such submanifolds cannot possible be invariant. I will then go on to discuss how (generalized) Seiberg-Witten equations might help cure these problems.

Title: Students’ graphing activities: Re-presentations of what?
Abstract:
Students’ representational activities are key to their mathematical development. Specifically, students’ representational activities in constructing displayed graphs can afford them the figurative material necessary to engage in and abstract mental operations. In this talk, I draw on Piagetian ideas to frame the sophistication of students’ ways of thinking for graphing. Namely, I illustrate distinctions between those ways of thinking dominated by sensorimotor experience and those ways of thinking dominated by the coordination of mental actions. Against the backdrop of these distinctions, I argue that we, as educators and researchers, need to broaden students’ representational experiences. Instructionally, doing so can afford students increased opportunities to construct productive and generative ways of thinking for mathematical ideas and concepts. In terms of research, broadening students’ representational experiences enables researchers to form more viable and detailed working hypotheses of students’ ways of thinking for graphing and related topics.

Title: Introduction to the arithmetic of modular forms. Part II.

Date: 10/10/2017

Time: 3:00 PM - 4:00 PM

Place: C304 Wells Hall

This is part of a 3 lecture series. The ultimate purpose of this lecture series is to explain a recent conjecture made jointly with Robert Pollack. The conjecture itself is about coarse p-adic invariants of modular forms called slopes, which are nothing other than the norms of the eigenvalues of a certain operator. By way of motivation, I will start by first discussing modular forms with a bias towards the arithmetic context of the Langlands program. The second talk will be reserved for exposing what one might call p-adic methods for modular forms. These have been around since the 70’s and 80’s and they are central to research on “p-adic Langlands” over the past 15 years. Finally, I will aim to state precisely the conjecture (called the ghost conjecture) Pollack and I have made and explain its numerical and theoretical evidence.

Title: Schur Symmetric Functions and Their Analogues, Part 1

Date: 10/10/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

I will continue to discuss some properties of the Schur symmetric functions
and then talk about some algebras related to the algebra of symmetric functions, such as noncommutative symmetric functions. In these related algebras I will discuss some analogues of the Schur symmetric functions and their properties.

Title: Stretching and Rotation Sets of Quasiconformal Maps

Date: 10/11/2017

Time: 4:10 PM - 5:00 PM

Place: C517 Wells Hall

Quasiconformal maps in the plane are orientation preserving homeomorphisms that satisfy certain distortion inequalities; infinitesimally, they map circles to ellipses of bounded eccentricity. Such maps have many useful geometric distortion properties, and yield a flexible and powerful generalization of conformal mappings. In this work, we study the singularities of these maps, in particular the sizes of the sets where a quasiconformal map can exhibit given stretching and rotation behavior. We improve results by Astala-Iwaniec-Prause-Saksman and Hitruhin to give examples of stretching and rotation sets with non-sigma-finite measure at the appropriate Hausdorff dimension.

In the second part, I generalize the above structures to the case of Riemann surfaces \Sigma_{g,s,n} -- R.s. with holes and decorated boundary cusps on hole boundaries. There, an interesting phenomenon occurs: (i) we have to consider generalized cluster transformations; (ii) we can establish 1-1 correspondence between (extended) shear coordinates and lambda-lengths, so we can investigate both Poisson and symplectic structures on the both sets of variables.

Title: Large scale geometry of asymptotically flat 3-manifolds

Date: 10/12/2017

Time: 11:00 AM - 12:00 PM

Place: C304 Wells Hall

Abstract: I will discuss recent work concerning the isoperimetric structure of asymptotically flat 3-manifolds and its relationship to the ADM and Hawking masses. This is joint work with M. Eichmair, Y. Shi, and H. Yu.

Every knot in the 3-sphere can be realized as a cross-section of some unknotted surface in the 4-sphere. For a given knot, the least genus of any of such surface is defined to be its double slice genus. Obviously twice the slice genus of a knot is a lower bound for its double slice genus. One really basic question is whether the double slice genus can be arbitrarily large compared to twice the slice genus. However, this was not answered due to the lack of lower bounds for the double slice genus. In this talk I will introduce a lower bound that can be used to answer this question.

Title: Galois groups in Enumerative Geometry and Applications

Date: 10/12/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

In 1870 Jordan explained how Galois theory can be applied
to problems from enumerative geometry, with the group encoding intrinsic structure of the problem. Earlier Hermite showed the equivalence of Galois groups with geometric monodromy groups, and in 1979 Harris initiated the modern study of Galois groups of enumerative problems. He posited that a Galois group should be `as large as possible' in that it will be the largest group preserving internal symmetry in the geometric problem.
I will describe this background and discuss some work
in a long-term project to compute, study, and use Galois
groups of geometric problems, including those that arise
in applications of algebraic geometry. A main focus is
to understand Galois groups in the Schubert calculus, a
well-understood class of geometric problems that has long
served as a laboratory for testing new ideas in enumerative
geometry.

Title: Schur Symmetric Functions and Their Analogues, Part 2

Date: 10/17/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

I will continue to discuss some properties of the Schur symmetric functions
and then talk about some algebras related to the algebra of symmetric functions, such as
noncommutative symmetric functions. In these related algebras I will discuss some analogues
of the Schur symmetric functions and their properties.

In this talk I present the general phenomenon of bubble in harmonic map case and extend it to Dirac-Harmonic map case.
Especially we discuss the difficulty to make bubble in Dirac-Harmonic case.

Title: Component Preserving Mutations and Maximal Green Sequences

Date: 10/19/2017

Time: 10:00 AM - 10:50 AM

Place: C304 Wells Hall

In this talk we will outline the study of maximal green sequences for cluster algebras. We will discuss new results which allow one to define component preserving mutations for quivers, and utilize them to create maximal green sequences by considering maximal green sequences for induced subquivers. This is ongoing work with John Machacek and three undergraduate researchers here at Michigan State University, Ethan Zewde, Evan Runberg, and Abe Yeck

Speaker: Elmas Irmak, Bowling Green State University

Title: Simplicial Maps of Complexes of Curves and Mapping Class Groups of Surfaces

Date: 10/19/2017

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

I will talk about recent developments on simplicial maps of complexes of curves on both orientable and nonorientable surfaces. I will also talk about joint work with Prof. Luis Paris. We prove that on a compact, connected, nonorientable surface of genus at least 5, any superinjective simplicial map from the two-sided curve complex to itself is induced by a homeomorphism that is unique up to isotopy. As an application we give classification of injective homomorphisms from finite index subgroups of the mapping class group to the whole group.

In this talk, I will present some connections between recent research in dynamical systems and the classical theory of elliptic curves and rational points. The main goal is to explain the role of dynamical stability and bifurcations in deducing arithmetic finiteness statements. I will focus on three examples: (1) the theorem of Mordell and Weil from the 1920s, presented from a dynamical point of view; (2) a recent result of Masser and Zannier about torsion points on elliptic curves, and (3) features of the Mandelbrot set.

The Mandelbrot set is one of the most famous objects in modern mathematics. We see images of it everywhere, but despite its popularity and decades of research, we still don't fully understand it. I will survey results about the Mandelbrot set, from its discovery to today.

Title: Uses of Neurocognitive Measures to Evaluate Cognitive Load During the Process of Proving

Date: 10/23/2017

Time: 12:00 PM - 1:00 PM

Place: 212 North Kedzie

A recent special issue of ZDM (June 2016) made the case for increasing the interdisciplinary collaboration between researchers in the fields of mathematics education and cognitive neuroscience. Specifically, Ansari and Lyons (2016) argued for increasing the ecological validity of the testing situations and specific [neurocognitive] tests used to measure mathematical processing” (pp. 379-380). To this end, Ansari and Lyons (2016) suggest that it would be useful to explore the use of lower-cost and less invasive neuroimaging methods such as Near Infrared Spectroscopy (NIRS). The study reported on in this talk serves as a “proof-of-concept” for the use of Frontal Near-Infrared Spectroscopy (fNIRS) to measure the level of cognitive load of the brain under mathematical processing. The talk will address the pros and cons of using neurocognitive measures, such as the fNIRS, to measure and examine the physiological stresses of the brain under the complex mathematical process of proving. It will also explore future studies designed to better understand the role and influence of neurocognitive structures (such as working memory capacity) in mathematical problem solving.

I will explain how to generate infinite families of rational balls from figure eight knot, which bound Rohlin invariant one Brieskorn homology spheres (a joint work with Kyle Larson). I will then discuss the question of when a contractible manifold fails to be a cork; this happens either if there is no exotic involution on its boundary, or it fails to be Stein. I will give examples of Stein non-corks, and non-Stein non-loose corks. I will discuss open questions of whether there are loose-corks that can not be corks, or if there are infinite order corks (this part is a joint work with Danny Ruberman).

Title: Beyond Aztec Castles and from Dungeons to Dragons

Date: 10/24/2017

Time: 10:00 AM - 10:50 AM

Place: C304 Wells Hall

In this talk I will discuss certain tessellations of the torus, known as brane tilings, and highlight how cluster algebras give rise to connections between combinatorics and physics. In particular, I will focus on algebraic and combinatorial formulas arising from quivers associated to del Pezzo surfaces and how such formulas allow us to investigate anew perfect matching problems of Propp from the turn of the century.

Title: Slopes of modular forms and the ghost conjecture.

Date: 10/24/2017

Time: 3:00 PM - 4:00 PM

Place: C304 Wells Hall

Abstract: In the previous two talks I discussed modular forms and their generalization p-adic modular forms. The space of p-adic modular forms is a p-adic Banach space equipped with an important operator U (the p-th Hecke operator). In this talk I will explain a conjecture recently formulated by Pollack and myself on the p-adic norms of eigenvalues of the U-operator acting on p-adic modular forms. Surprisingly, our description of these norms is completely elementary. If time permits I will discuss some of the evidence we have for our conjecture and its natural generalizations.

Title: A Combinatorial Formula for Birational Rowmotion on Rectangular Posets

Date: 10/24/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

We investigate iterated applications of a dynamic known as birational rowmotion on rectangular posets, i.e. the product of two chains. We obtain a combinatorial formula stated in terms of the weighted enumeration of non-intersecting lattice paths. This work began at the 2015 AIM (American Institute of Mathematics) Workshop on Dynamical Algebraic Combinatorics. We thank AIM for its hospitality.

Title: Arithmetic of Dehn surgery points and Azumaya algebras

Date: 10/26/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Associated to a finite volume hyperbolic 3-manifold is a
number field and a quaternion algebra over that number field. Closed hyperbolic 3-manifolds arising from Dehn surgeries on a hyperbolic knot complement provide a family of number fields and quaternion algebras that can be viewed as "varying" over a certain curve component (the so-called canonical component) of the SL(2,C)-character variety of the knot group. This talk will give examples of different behavior and survey recent work on how the varying behavior can be explained using the language of Azumaya algebras over the canonical curve.

Speaker: Prof. Fatih Celiker, Department of Mathematics, Wayne State University

Title: Novel nonlocal operators in arbitrary dimension enforcing local boundary conditions

Date: 10/27/2017

Time: 4:10 PM - 5:00 PM

Place: C100 Wells Hall

Abstract: We present novel nonlocal governing operators in 2D/3D for
wave propagation and diffusion that enforce local boundary conditions
(BC). The main ingredients are periodic, antiperiodic, and mixed
extensions of kernel functions together with even and odd parts of
bivariate functions. We present all possible 36 different types of BC
in 2D which include pure and mixed combinations of Neumann, Dirichlet,
periodic, and antiperiodic BC. Our construction is systematic and
easy to follow. We provide numerical experiments that validate our
theoretical findings. We also compare the solutions of the classical
wave and heat equations to their nonlocal counterparts.

Title: Geometric Structures on Representation Varieties

Date: 10/30/2017

Time: 3:00 PM - 3:50 PM

Place: C304 Wells Hall

We define varieties which parameterize isomorphism classes of finite-dimensional representations of a group or an algebra, and then discuss symplectic and Poisson structures on the algebra of functions of such a variety.

Title: Constructing maximal green sequences from subquivers

Date: 10/31/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Quiver mutation was first studied in the creation of cluster algebras by Fomin and Zelevinsky to study total positivity. Since then it has shown applications to a wide variety of other areas of mathematics. At its core is a way of taking one directed graph and mutating at a vertex to produce a new direct graph. In this talk we will introduce the process of quiver mutation and discuss special sequences of mutation called maximal green sequences. Time permitting we will discuss joint work with John Machacek, Evan Runberg, Abe Yeck, and Ethan Zewde which shows how one can construct maximal green sequences from induced subquivers.